Massimo Nespolo's research themes

Crystallography of mica polytypes and twins

PTST98

PTST98 (PolyType STacking) is a simple program
that computes the
Periodic
Intensity Distribution
(PID) function of mica polytypes.

Mica polytypes are built by stacking a unit layer
(commonly indicated as «M layer» through one or more generating operations:
these are local symmetry operations (space-groupoid operations) valid in the
crystal subspace corresponding to a pair of layers, but not valid in the whole
crystal space. Successive layers are differently oriented, and the relative
rotations about
c
* are
n
x 60°
(
n
= 0-5). These rotations are nothing else than a
description
of the stacking mode in micas: they are not
generating
operations from either the geometrical or the crystal-growth viewpoint.

The number of polytypes increases
dramatically with the number of layers (
Mogami
et al.
,
1978
;
McLarnan, 1981
) and the identification of the stacking sequence of a long-period mica
polytype requires thus an indirect procedure which exploits the periodicity in
reciprocal space. This simplified procedure was introduced by
Takeda (1967)
under of the name of Periodic Intensity Distribution (PID). The PID is an
approximant of the Fourier transform of the stacking sequence that can be
obtained in a simple way from the diffraction intensities.

The Fourier transform of a polytype (
GN,
where
N
is the number of layers) is given by the Fourier transform of the
stacking sequence, modulated by the Fourier transform of the layer (
Gj
):

where
tx,j,
ty,j,
tz,j
are the (
x, y, z
)
components of the stacking vector relating the
j
-th and the (
j
+1)-th
layers. When the shifts between the building layers are rational
and the rotations belong to the symmetry of the layer(s), their Fourier
transform (
Gj
), which is a continuous function in the
direction of lacking periodicity, can be extracted from the expression of the
structure factor
GN.
GN
takes thus the
simple form of the product of the layer transform and of the stacking sequence
transform. The second term expresses the periodicity in reciprocal space
appearing when a structure is built by translation of subunits. This is the case
of polytypes of binary compounds like SiC and ZnS.

In micas, the M layers are
instead related by rotations belonging not to the layer symmetry but to the
idealised symmetry of the O
b
plane (with the obvious exception
of the 1
M
polytype) and the same simplification is in principle not
possible. However the Fourier transform of the M layer in the six possible
orientations is almost unmodified in a
subspace
of the reciprocal space
(
Takeda, 1967
). By removing the modulating effect of the layer, the
approximated
Fourier transform of the stacking sequence is obtained. This is known as
Periodic
Intensity Distribution
(
PID
) function. Comparison of calculated and observed PID
values along lattice rows parallel to
c
* with
k
≠ 3
n
(
non-family rows:
Ďurovič
et al.
,
1984
;
X
rows:
Nespolo
et al.
, 2000
) is in principle
sufficient to identify any mica polytype (
Takeda and Ross,
1995
;
Nespolo
et al.
1999
).

The
PTST98 program uses the
RTW rotational symbols
(
Ross
et al.
, 1966
) of a stacking candidate to compute the PID function.
Instructions and examples are included in the downloadable ZIP file.